Disconnected Cuts in Claw-free Graphs

Authors Barnaby Martin, Daniël Paulusma, Erik Jan van Leeuwen

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Author Details

Barnaby Martin
  • Department of Computer Science, Durham University, Durham, UK
Daniël Paulusma
  • Department of Computer Science, Durham University, Durham, UK
Erik Jan van Leeuwen
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands

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Barnaby Martin, Daniël Paulusma, and Erik Jan van Leeuwen. Disconnected Cuts in Claw-free Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 61:1-61:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The corresponding decision problem is called Disconnected Cut. It is known that Disconnected Cut is NP-hard on general graphs, while polynomial-time algorithms exist for several graph classes. However, the complexity of the problem on claw-free graphs remained an open question. Its connection to the complexity of the problem to contract a claw-free graph to the 4-vertex cycle C_4 led Ito et al. (TCS 2011) to explicitly ask to resolve this open question. We prove that Disconnected Cut is polynomial-time solvable on claw-free graphs, answering the question of Ito et al. The basis for our result is a decomposition theorem for claw-free graphs of diameter 2, which we believe is of independent interest and builds on the research line initiated by Chudnovsky and Seymour (JCTB 2007-2012) and Hermelin et al. (ICALP 2011). On our way to exploit this decomposition theorem, we characterize how disconnected cuts interact with certain cobipartite subgraphs, and prove two further algorithmic results, namely that Disconnected Cut is polynomial-time solvable on circular-arc graphs and line graphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • disconnected cut
  • surjective homomorphism
  • biclique cover
  • claw-freeness


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